A particularly important special case of setup (REF ) is training an ML model to predict what would happen in (quantum) physical experiments [1]}.
Such experiments might explore, for instance, the outcome of a reaction in quantum chemistry [2]}, ground state properties of a novel molecule or material [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, or the behavior of neutral atoms in an analog quantum simulator [10]}, [11]}, [12]}. In these cases, the input \(x\)  subsumes parameters that characterize the process, e.g., chemicals involved in the reaction, a description of the molecule, or the intensity of lasers that control the neutral atoms. The map \(\operatorname{{\mathcal {E}}}\)  characterizes a (potentially very complicated) quantum evolution happening in the lab. Depending on the parameter \(x\) , it produces the quantum state \(\operatorname{{\mathcal {E}}}(\vert x \rangle \! \langle x \vert )\) .
Finally, the experimentalist measures a certain observable \(O\)  at the end of the experiment.
The goal is to predict the measurement outcome for new physical experiments, with different values of \(x\)  than those encountered during training.
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